Pricing a Zero-Coupon Bond
Pricing a Zero-Coupon Bond
A zero-coupon bond pays no interim coupons. Its price is simply the face value discounted at the market yield to maturity. The formula is the simplest, but the behavior is the most extreme.
Key takeaways
- A zero-coupon bond's price is P = FV / (1 + y)^n, where n is the number of years to maturity and y is the annual yield.
- Zeros are purchased at a deep discount and redeemed at par at maturity; the difference is the return.
- The absence of coupon payments makes zeros extremely price-sensitive to yield changes; they are the most volatile bonds of any given maturity.
- Zero-coupon bonds are created by stripping conventional Treasury bonds or by issuance of zero-coupon Treasury debt.
- Zeros are useful for matching specific long-term liabilities (education costs, retirement, pension obligations) because the maturity and return are certain.
The zero-coupon pricing formula
For a zero-coupon bond, the pricing formula simplifies to a single term:
P = \frac{FV}{(1 + y)^n}
where:
- P = Bond price (today)
- FV = Face value (the amount repaid at maturity)
- y = Annual yield to maturity
- n = Years to maturity
This is the pure present-value formula: the face value is discounted backward by n years at rate y. There are no coupons to discount separately; the entire return comes from the difference between purchase price and face value.
Example: Pricing a 10-year Treasury zero
Suppose the 10-year Treasury zero-coupon yield is 4.5%. A zero with face value $1,000 matures in 10 years. The price is:
P = \frac{1,000}{(1.045)^{10}} = \frac{1,000}{1.5531} = \$643.80
The investor buys the bond for $643.80 and receives $1,000 at maturity 10 years later. The return is $1,000 - $643.80 = $356.20 in total profit, or equivalently, 4.5% annualized over 10 years.
Now suppose yields rise to 5%. The new price is:
P = \frac{1,000}{(1.05)^{10}} = \frac{1,000}{1.6289} = \$614.46
Price change: $614.46 - $643.80 = -$29.34, or -4.6% for a 50 basis point yield increase.
If yields fall to 4%, the price becomes:
P = \frac{1,000}{(1.04)^{10}} = \frac{1,000}{1.4802} = \$675.56
Price change: $675.56 - $643.80 = $31.76, or +4.9% for a 50 basis point yield decrease.
Notice the symmetry (not exact, but close): a 50 bp rise causes a 4.6% decline, and a 50 bp fall causes a 4.9% gain. This is the extreme price sensitivity of long-maturity zeros.
Comparing zeros to coupon bonds of the same maturity
We can compare the 10-year zero above to a 10-year Treasury with a 4.5% coupon, both priced at a 4.5% yield:
10-year coupon bond (4.5% coupon, 4.5% yield):
- Price = $1,000 (par)
- At 5% yield: Price ≈ $955
- Price change: -4.5%
10-year zero (4.5% yield):
- Price = $643.80
- At 5% yield: Price ≈ $614.46
- Price change: -4.6%
In this case, the zero's price change is very similar to the coupon bond's (-4.6% vs. -4.5%). But this is because we started at par and a matched yield. The key difference is that the zero achieved this match at a much lower starting price. The zero's percentage volatility is similar or slightly higher, but its dollar volatility is smaller (because the base price is smaller).
Now compare with a lower-coupon 10-year bond (1% coupon, 4.5% yield):
10-year 1% coupon bond at 4.5% yield:
- Price ≈ $795
- At 5% yield: Price ≈ $759
- Price change: -4.5%
Again, the percentage change is similar. The lesson: price sensitivity (in percentage terms) depends on duration, which is determined by maturity, coupon, and yield. A zero with 10-year maturity has similar duration to a coupon bond of similar maturity, so they have similar percentage price volatility.
However, zeros can be more extreme in longer maturities. A 30-year zero is extraordinarily volatile, with a duration around 30 years. A 30-year coupon bond has a duration of maybe 18–20 years. The zero's greater duration makes it more volatile.
Real-world use: Treasury Strips
The US Treasury does not issue zero-coupon securities directly, but they are created through a process called stripping. Large financial institutions take conventional Treasury bonds and separate the coupons from the principal. Each coupon becomes a mini-zero, maturing on the coupon date. The principal becomes a final zero, maturing at the bond's stated maturity. These stripped securities are known as Treasury Strips.
For example, a 30-year Treasury Bond pays coupons for 60 semi-annual periods and then returns principal. Stripping it produces:
- 60 mini-zeros (one for each coupon payment)
- 1 large zero (the principal at the final maturity)
These Strips trade in the secondary market, and their prices follow the zero-coupon formula. A trader buying a 15-year Strips zero expects to hold it for 15 years, receiving $1,000 at maturity and nothing in between.
Practical applications: Liability matching
Zeros are particularly useful for investors or institutions that need to match specific future liabilities. A parent saving for a child's college education 18 years away can purchase an 18-year zero-coupon bond (or a portfolio of zeros) and know with certainty that it will be worth $1,000 (or the required amount) at the desired maturity. No coupon reinvestment risk, no interim price volatility (though the bond still experiences market price changes that matter if sold before maturity), and a fixed terminal value.
Similarly, pension plans that need to fund specific obligations 10, 20, or 30 years in the future use zeros to eliminate reinvestment risk. A pension fund that buys 20-year zeros to cover obligations 20 years from now will receive exactly the principal needed, regardless of interim coupon rates. This is called immunization.
Tax considerations
One drawback of zeros is tax inefficiency. Although you receive no cash until maturity, the IRS treats the annual discount accretion as taxable income each year. An investor buying a 10-year zero at $643.80 is taxed annually on the implied interest income, even though no cash is received. This is why zeros are best held in tax-deferred accounts (IRAs, 401(k)s) rather than taxable accounts.
Flowchart
Next
Zeros represent one extreme: maximum price sensitivity for a given maturity. In practice, most investors hold coupon-paying bonds in funds or individual securities. Understanding how bond prices evolve over time—from purchase to maturity, through yields rising and falling—requires visualizing the price-time relationship. In the next article, we explore graphs and charts that show how a bond's price behaves as maturity approaches.